What the sacred geometry folk call a merkaba is what the rest of us call the Stella Octangula, except generally without the rotation you refer to. The full version of Stella provides over a hundred compounds, of which the Stella Octangula is just the simplest These are not available in the demo though, but in the full version the Stella Octangula can be found at "Stella Library->Compounds->Tetrahedra 2".

If it wasn't already in the library, there are several ways it could be created in Great Stella or Stella4D. One is to start with an octahedron and stellate it. Another is to start with a cube and facet it. Actually you can do both these things in the demo versions. Here's a screenshot taken from the demo:

To me the sense of sacredness people may feel, whether it comes from geometry, crystals, or ancient texts etc, is just one convenient way people interpret a sense of awe and embrace the unknown. I'm personally quite a sceptic and an atheist, so I might find beauty in geometry, but recognise that this is a subjective impression rather than representing an objective reality. But it's fun to discuss such concepts.

I certainly don't plan to delete posts just because they have views that differ from my own though, so feel free to keep posting

Nice job Ulrich. I'm not sure how you did it either! You wouldn't have been able to see this animation in Stella though, and must have created each frame in the animation as a separate model and then strung them together, right? (Just don't want to mislead katie in this regard). It looks great though

Someday I'll get around to adding a compound generator, and then it should be possible to view this rotating compound in Stella (I think).

can you make it rotate, the top spinning opposite to the bottom?

I'm not sure what's being asked for here.

And Katie you're right that rotation is about two axes (according to horizontal or vertical mouse movement). It would indeed be very useful to be able to lock rotation to a specific rotational symmetry axis or constrain it in some other way. I wonder what the best interface for this would be? Maybe a rotation-lock button, after which the user clicks on the symmetry axis (or other entities) which they want to lock the rotation to. Then any left-drag would rotate solely around that axis. However, I imagine it would also be nice to tumble in the normal way while still rotating around that axis. So it would be nice to have access to both forms of rotation at once. Not sure of the best interface for this. Any thoughts?

It is a series of 20 images which I exported from stella. As Robert says, I created 20 different models, starting from a 60-fold prism. By faceting, I got 20 3-fold prisms from it which I trimmed to the proper height which is the distance of two parallel planes in the stella octangula. I did this with the augmentation feature, augmenting a 20-prisms-compound to each one of two triangles of the stella octangula, lying in parallel planes. Thus I got the vertex points for faceting a new 20-prisms-compound with the proper height. I put this into the memory and excavated a tetrahedron from one top triangle and another one from the bottom triangle of the neighboured prism. I orientated this like looking at a 2-fold axis. Then I removed the 20-prisms-compound by excavation with the model in the memory. After recolouring one of the two remaining tetrahedra (carefully, without moving it!) and hitting "f" to fit the model in the screen, I exported the first picture with a solution 300x300. Then I retrieved the 20-prisms-compound from the memory and repeated this procedure 19 times, going one prism to the left and one to the right in each step. Here the diagnostic mode "show vertex numbers" is very helpful. It has to be deactivated each time before creating the image. Then, the images must be strung together with an animation program.
Rotating the top vertex opposite the bottom one can be realized in a similar way by putting a stella octangula into a cube.

yes, it would be nice to lock one axis. perhaps if the user could click on the axis they wanted to lock, by choosing one of the colored pins that stick out of the vertices and faces when you push the little yellow menu wheel called "show symmetry axis." those little pins are really very handy and useful.

Rob, i really hope you can help with this: "can you make it rotate, the top spinning opposite to the bottom? / I'm not sure what's being asked for here."

for a merkaba (sacred tetrahedron), or the one i like even better, the octahedron, to function as it is supposed to -- then the top half of the figure must be set to spinning or turning on the vertical axis, and the bottom half must be set to spinning on the same vertical axis but in the opposite rotation.

deasil is clockwise and withershins or widdershins in counterclockwise with respect to the single axis of rotation joining the entire figure.

so, for instance, if we could get two johnson solids of the square pyramid configuration placed on the same screen in stella or stella4d and with the square "bottoms" touching, and then set each one to spinning in the opposite direction, then you would have the starting position of a sacred octahedron, (before they interpenetrate).

and if i might ask another question. what is a "dual" exactly?

and what does it mean to stellate a particular polyhedron?

and finally why does the "dual" of the johnson solid-square pyramid show up as two very different sizes? if they were equal sizes, and interpenetrated like they are, and then each spinning opposite, this would be perfect.

all the pentagram figures are quite enticing. i really want to work on these also.

My goodness, Ulrich. that is quite a bit of work you did, and the figure is lovely with a great choice of colors.

this is a very nice static representation of a functioning merkaba, which is often shown symbolically as a star of david (two equilateral triangles equally interpenetrated).

i like the way you have implied and portrayed the motion of opposite spin in a static figure, that is quite ingenious.

and what an incredible amount of work. you must understand the program very well.

Thank you so much, Ulrich.

so here are you meaning that an opposite spin of each tetragon can be shown in continuous motion, within the stella program so as to make a dynamic moving image of a merkaba:
"Rotating the top vertex opposite the bottom one can be realized in a similar way by putting a stella octangula into a cube."

3katie3 wrote:yes, it would be nice to lock one axis. perhaps if the user could click on the axis they wanted to lock, by choosing one of the colored pins that stick out of the vertices and faces

Well yes, but I was thinking further down the track. If left-drag would perform the locked rotation, then you would not also be able to tumble the model as usual. I think it would be nice to be able to tumble while the locked rotation continues on its own.

Rob, i really hope you can help with this: "can you make it rotate, the top spinning opposite to the bottom? / I'm not sure what's being asked for here."

for a merkaba (sacred tetrahedron), or the one i like even better, the octahedron, to function as it is supposed to -- then the top half of the figure must be set to spinning or turning on the vertical axis, and the bottom half must be set to spinning on the same vertical axis but in the opposite rotation.

That's what I thought you meant, but isn't this what Ulrich already did?

and finally why does the "dual" of the johnson solid-square pyramid show up as two very different sizes?

They are not so much different sizes as different shapes. The issue comes down to the choice of centre for reciprocation (used to find the dual). A square pyramid doesn't have an obvious centre, and the choice leads to distorted versions of the dual. Finding a "canonical" centre for reciprocation remains an unsolved problem for geometry at the moment.

Sorry, that explanation probably doesn't mean much to you, but it's a bit hard to explain in simple terms. Basically it has to do with the square pyramid not having a single obvious centre.

3Skatie3 wrote:for a merkaba (sacred tetrahedron), or the one i like even better, the octahedron, to function as it is supposed to -- then the top half of the figure must be set to spinning or turning on the vertical axis, and the bottom half must be set to spinning on the same vertical axis but in the opposite rotation.

You can have it rotate by the 4-fold axis. Then it looks like this:

or like this

or you may look down the 3-fold axis:

I made the first one of these by putting a stella octangula into a cube and starting with an 80-fold prism.
Ulrich

Last edited by Ulrich on Fri Sep 12, 2008 1:01 pm, edited 3 times in total.